Postulation and gonality for projective curves

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

E DOARDO B ALLICO

Postulation and gonality for projective curves

Rendiconti del Seminario Matematico della Università di Padova, tome 80 (1988), p. 127-137

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Gonality Projective

EDOARDO BALLICO

(*)

We are interested in the

interplay

^{between intrinsic and}

projective properties

^{of curves. In}

particular

^{we are} interested in the

postulation

of

general k-gonal

^{curves. A smooth curve Y c PN is said to be canonical}

if

1".1 Ky.

^{A curve} Z c PN is said to have maximal rank if the restrictions maps

r,,,(k): HO(PN,

^-

HO(Z, c~~(k)~

^{have maximal}

rank for all

integers

^{k. In}

[4]

^{it was}

proved

^that for every N &#x3E;

3, g &#x3E; N,

^a

general non-degenerate

^{canonical curve} with genus g in PN has maximal rank.

Here we prove the

following

^results

(over C).

THEOREM 1. For all

integers

^N &#x3E;

3, g

&#x3E;

N,

^a

general trigonal (resp. bielliptic) non-degenerate

^{canonical curve of} genus g ⁱⁿ PN has maximal rank.

THEOREM 2. For all

integers N, d,

g,

with g

&#x3E; N &#x3E;

3,

^d &#x3E;

2g,

^a

general embedding

^of

degree d

^{in PN of a}

general hyperelliptic

^curve

of genus g has maximal rank.

The

proofs

of theorem 1 and theorem 2 is modulo a

smoothing

result

given in § 1,

almost the same that the

proofs

ⁱⁿ

[4];

ⁱⁿ

parti-

cular

in §

^{3 we omit the} details which can be found in

[4]

^or

[3].

^For

§

^{1 we use the}

theory

^of admissible

coverings ([7])

which is very useful to obtain results about

general k-gonal

^curves

by degeneration

^tech-

niques.

^{The inductive method}

of § 2, § 3, (the

^Horace’s

method)

^was

introduced in

[8].

(*) Indirizzo dell’A.:

Dipartimento

^di Matematica, Università di Trento, 38050 Povo

(TN), Italy.

In §

^{4 we show}

that, by

^the

theory

^of admissible covers, the results

proved

ⁱⁿ

[6]

^{about the minimal} free resolution of

general

^canonical

curves are true

(with

^{the same}

proof)

^for

general k-gonal

^{curves for}

suitable k.

0. Notations.

Let V be a

variety (over C)

^{and S a closed subscheme of}

V; 3s,v

is the ideal sheaf of S in V and

Ns,v

its normal sheaf

(or

^normal

bundle).

Assume that we have fixed an

embedding

of V into

Pk,

^{so that}

0v(t)

and are defined. Then

rs,v(t): HO(V,

^{--~ is the}

restriction map. If V =

Pk,

^we write often

rs,k(t), Ns,k

^instead

of

rs,v(t), IS,V .

⁰

A curve T c PNis called a bamboo of

degree

d if it is

reduced,

^con-

nected,

^{with at most nodes as}

singularities, deg (T)

⁼

d,

^{its irre-}

ducible

components

^are

lines,

and each line in T intersects at most two other irreducible

components

^of

T; equivalently,

^we may order the lines

.L1,

... ,

Z~

^{of T so that iff and}

only

^if

A

connected,

reduced curve X c

PN, deg (X)

⁼

2d,

^{..g with}

only

^or-

dinary

^{nodes as}

singularities,

is called a chain of d conics if its irre- dicible

components C1,

... ,

C,

^are

conics, ⁰

^{if and}

only

^if

I i - j 2,

^card

(Ci

ⁿ

Ci+,)

^{= 2 if 0 i d. Sometimes we will}

allow the

reducibility

^{of some} of the conics

Ci

i in a chain of

conics ;

if

Ci

^is

reducible,

^{we assume} that every line of

Ci

intersects the

adjacent

conics. A line

(resp.

^a

conic)

^{of a bamboo}

(resp.

^{chain of}

conics)

^T

is called final if it intersects at most another irreducible compo- nent of T.

We will write

( (a ; b))

for the binomial

coefficient ;

^thus

( (a ; b )) : = : _ (a ! ) / ((a - b ) ! b ! ) .

^A

^triple

^of

integers (d, g ; N)

^with

d &#x3E; g + N,

N &#x3E;

2, g ~ 0

has critical value k is the first

integer

t &#x3E; 0 such that td

+

^1- gc

((N ⁺ t; N)).

^{We define}

integers r(k,

g,

N), q(k,

g,

N) by

the

following

^relations:

A smooth curve T of

degree d

and genus g in PNwith

r(k - 1,

g,

N)

and

hl(T,

²

^k,

^has critical value 1~

(i.e.

(d, g; N)

^{has critical value}

1~) ;

T has maximal rank if and

only

^if

-

1 )

^is

injective

^and

r,,,(k)

^is

surjective (Castelnuovo-Mumford’s

Lemma).

We define

integers c(k, N), e(k, N) by

^the

following

relations :

A chain F of

c(k, N)

conics in PN has critical value

1~;

^{it has maximal}

rank if and

only

^if

rF,N(k

^-

¹⁾

^is

injective

^and

HO(PN,

⁼⁼

e(k, N).

Define

integers y(k, N), k

&#x3E;

0,

N &#x3E;

2,

^{in the}

following

way. Set

y(1, N) := c(l, ^N)

^{and assume defined}

y(k - 1, N).

^Set

y(k, N) _ e(l~ - 1, N), e

^{= 0} otherwise. Hence

N) &#x3E; y(k, N)

&#x3E;

N) -

^k.

Define

integers N), j(k, N) by

^the

following

relations :

A canonical curve C of

degree

d in PN has critical value k &#x3E; 1 if and

only

^if

2x(k - 1, N)+2d2x).

^Set

y’ (k, N) : = y(k, N) - - [(N + 5)/3]-1.

A finite subset S c Pk is said to be in Linear

general position

^if

every subset W of S spans ^a linear space of dimension min

(k,

card

(W) - 1).

^..

A

bielliptic

^{curve is a}

smooth, connected, complete

^{curve with a}

degree

^two

morphism

^{onto an}

elliptic

^curve.

Let Y = A u B c

P3,

⁷ A chain of

conics, B bamboo, A

^intersect-

ing

^{B at a}

unique point, P,

^and

quasi-transversally,

^P

belonging

^to

a final line of B and a final conic of A. An irreducible

component

^{of Y}

is called free if it intersects

only

another irreducible

component

^{of Y.}

1. A

smooth,

^{connected curve Ec Pn} is called canonical if Let

C(g, n)

^{be the closure in Hilb}

(pn)

^{of the set of}

smooth canonical curves of genus g in

Pn ,

^and

C(g,

n,

k) (resp. ^C(g,

^n;

biel))

^{the closure in Hilb of the set} of smooth canonical curves

of genus g ⁱⁿ

Pn,

^{which are}

k-gonal (resp. bielliptic)

^{as abstract curves.}

PROPOSITION 1.1. Fix a smooth canonical

trigonal

^{curve C c}

pn, deg (C)

⁼

2g - 2,

²

points A,

^{B in the same fiber of on}

C,

^{and a}

chain D of r conics in

Pn,

^{7 D}

intersecting

^C

quasi-transversally

^and

exactly

^{at A and}

B,

^{A and B}

belonging

^{to a} final conic of D. Then

C u D c- C(g + r, n, 3).

PROOF.

By

definition

C(g +

r, n,

3)

is closed in Hilb

(Pn).

^Hence

we may ^assume

C, A,

^D

general.

We know that C u D E

C(g, n)

([5], ^{[4], §} 2~ ~

First assume r ⁼ 1.

Taking

^a

projection,

^we may ^{assume n =} g, C V D

spanning P",

^C

spanning

^a

hyperplane M,

^and

hl(C, NO,M)

^{= 0.}

We know that

hl(C

^U

D,

⁰

([4], proof

^of

2.1),

^{hence C u D}

is a smooth

point

^{of Hilb}

(Pn).

^{Since C u D is}

semi-stable,

^{we have a}

morphism h

^{from a}

neighborhood

^{of C u D in Hilb}

(P")

to the moduli scheme of stable curves of genus

g + 1

^{such that}

h(C

^u

D)

^is

the curve C’ obtained from C

pinching together

^the

points ~,

^B.

By

the

generality of C, A,

^we may ^assume Aut

( C’ )

^_

{1},

^{i.e. C’ is a}

smooth

point

^of

Mg+1.

^{To obtain 1.1 for r =}

1,

it is sufficient to check that h is

flat,

hence open, at C ^U D.

By

the smoothness of Hilb

( lPn)

and

Mug+1

^{at the}

corresponding points,

^it is sufficient to check that the fiber has the

right

^dimension

n2 +

^{2n = dim}

(Aut ( l~n))

^{in a}

neiborhood of C u D. A

priori

^{near C U}

Dh-’(C’)

contains either curves

abstractly isomorphic

^{to C’}

(i.e.

irreducible canonical stable

curves)

or curves

isomorphic

^{to C U} D. The first

type

^{of curves} has dimension

n2 -i--

^{2n. Since Pic}

(C

^u

D)

^{has a} 1-dimensional

non-compact factor,

we see

easily that,

^{up to}

projective transformations,

^{there is}

exactly

a one dimensional

family

^{of curves C" U} C V D. However for any ²

triples

⁼

1, 2, 3,

of distinct

points

^of

D",

^{there is}

with

Fi

for every i. Hence the stabilizer of any C" U D" in Aut is

one-dimensional, concluding

^the

proof

^of

the case r = 1.

By

^{induction on r, if r} &#x3E; 1 it is sufficient to prove the

following

^claim

stronger

^{than the case r =}

1 just

proven.

Claim. Assume r ⁼ 1 and fix 2

general points

^of

D;

^then

there is a flat

family X - T,

T smooth irreducible affine curve, Xc

T X P",

^{with =}

.~t smooth,

canonical and

trigonal

^for

t E

T,

^{t ~}

0,

^{and a}

family

^{mt of}

3-coverings,

^mt:

^{Xi -}

PI such that is the

given g3 , molD

^sends

A, B

^{to one}

point

of Pi and

E, F

to another

point

^{of l~l.}

By

^the

theory

of admissible

coverings ([7])

^{there is a}

^morphism

b : with

b(O)

⁼

C’, b(t)

^{a smooth}

3-gonal

^{curve for 0.}

By

the first

part

^{of the}

proof

^we may ^assume

Xt = b(t). By [7], proof

^{of we the} existence of a

family

^{’int: -}

t E

which,

^{as t} goes ^to

0,

^{tends to an} admissible

covering

^mo

with ⁼

mo (B ),

⁼

Taking

^a suitable fiber

product,

we obtain the claim.

PROPOSITION 1.2. Fix a canonical

bielliptic

^{curve C c}

pn, deg (C)

^_

- 2g - 2,

²

points A,

^{B on} C with the same

image

^{under the 2 to 1}

map of C to ^an

elliptic

curve, and ^a chain D of r conics in D inter-

secting

^C

quasi-transversally

^and

exactly

^{at A and}

B,

^{A and B}

belong- ing

^{to a final conic} of D. Then C U D E

C(g +

r, n;

biel).

PROOF. The

proof

^{of 1.1} works with two minor twists. We use

the notations introduced in the

proof

^{of 1.1. Since Aut}

(C)

^and

Aut

( C’ )

^{is not}

trivial,

^{is not smooth at} C’. Instead of

we may however ^use

(over C)

the Kuranishi local deformation space

ov C’

(or

^{a suitable}

rigidification

^of Instead of admissible cover-

ings

^of

~1,

^{we have to use} admissible

2-coverings

^{of curves} of arithmetic genus ^1. Since these

coverings

^are

cyclic,

^{there is no} need here of a

general theory.

^{Take a}

2-covering

^c:

C --~ Z,

^Z

elliptic

curve, with

c(A)

⁼

c(B) (hence

^{c not ramified at}

A, B )

^{and a}

2-covering

^{d :}

with

d(A)

⁼

d(B) (hence

unramified at A and

B).

^{Take as Z U} ]p&#x3E;1 the

glueing

of Z and P’

along c((A)

^and

d(.E).

^Then c, d induce ^{a 2-cover-}

ing u :

^{C U D - Z U PB Let} ... , a2g be the ramification

points of u,

with a, ^E Z if and

only

^if

i o 2g -

^2. Take any flat

family

^{s : W -}

T,

with

Wo

^{= Z U}

P’7 W,

^smooth

elliptic

^{for t ~}

0, and,

^{in an etale}

neighborhood

^of

0,

any

2g

sections sl, ..., s2g of s with

8,(0)

^{= ai:}

The divisor

-~-

^...

+ s29 (t )

^on

Wt

^{induces a}

cyclic 2-covering

^which

tends to u when t goes ^to 0. 0

Let

Z(d, g,

n,

k) (resp. Z(d, g,

^n;

biel))

be the closure in Hilb of the set of

smooth, connected, k-gonal (resp. bielliptic)

^{curves of}

degree d,

genus g, and with ^non

special hyperplane

^section.

Z(d,

g, n,

k)

and

Z(d,

g, n;

biel)

^are irreducible

(and

^not

empty

^if

d ~ g ⁺ n).

^We

have the

following

^result.

PROPOSITION 1.3. Fix C c

Pn,

^{C E}

Z(d, g,

^n,

k) (resp. ^{Z(d, g,}

^n;

biel)),

2 smooth

points A,

^{B on} C with the same

image

^{under an admissible}

covering

^of

degree k (resp.

^{a 2-to-1}

covering

^{of a curve of genus}

¹⁾

of

C,

^{and a} chain D of r conics in

Pn,

^D

intersecting

^C

quasi-trans- versally,

^and

exactly

^at

A, B,

^{A and B}

belonging

^{to the same final}

conic of D. Then C U D E

Z(d + 2r, g +

r, n,

k) (resp. ^{Z(d + 2r, g + r,}

conic of D. Then C u D E

Z(d + 2r, g ⁺

^{r, n,}

k) (resp. ^{Z(d + 2r,}

g +

r, n ;

biel)) .

PROOF. It is much easier than the

proof

^of

1.1,

^{1.2. Take any}

flat

family

of admissible

covering, (W - T,

^{U -}

T,

^{W’ --~}

U)

^with

Wo

^{= C U}

D, Uo

^{= Z U}

Pi,

^{= 0}

(resp. 1)

^{and d}

+

^{2r sections} ~,vi of T with

+ ... + Wà+2r(0)

^an

hyperplane

section of

Wo.

^Embedd

W, using + ... + and,

if necessary,

project

^{the re-}

sult in

We shall use often the

following

^fact

([9], [2], 3.5).

^{Fix a smooth}

curve C c ₇

deg ( C)

⁼

d,

^{and a}

smooth,

^{connected curve}

D, deg (D)

^{= r, D}

intersecting quasi-transversally

^{C and}

exactly

^{at a}

point.

Assume D rational and that the

hyperplane

section of C is

non-special.

^{Then C U D is a limit of}

embeddings

^of

degree

^d

+ r

of C into P".

2. In this section we construct curves Y = Z U T c

P3 ,

^with

Z n T =

0,

^{Z chain of}

conics,

^T

bamboo,

^{and with}

good postulation.

This construction will be used in the next section to prove theo-

rems

1,

^{2 in P4.}

LEMMA 2.1. Fix

non-negative integers a, b, r, s,

^{and a smooth}

quadric Q

^{in P3 .} Assume either

(i) a

^{= b} &#x3E;

2,

^or

(ii) a

&#x3E; b &#x3E; 1. Then there is

( Y, Z, T)

^with

Z r1 T = ~, Z

chain of r

conics,

T bamboo of

degree s,

^dim

( Y

^r1

Q)==0y

^and

PROOF. - From now on, ^{we assume s =}

0,

^the

general

^case

being

similar

(or

^use

^[1], 6.2).

^{If 8 =}

^0,

^{it is} sufficient to prove ^2.1 when

(a + 1 ) (b + 1) -

^{4 4r}

(a + 1 ) (b + 1) +

^4.

By

the properness of Hilb

(P3),

for any u and any

plane

^{.g there is a scheme}

W,

^with

Wred C H,

^with

only ordinary

^double

points,

^{W reduce outside}

the

singular

^{locus of W limit of a}

family

of chains of u conics. Just to fix the

notations,

we assume a

-~- b -~-

^{1 = 3 mod}

(4),

the

remaining

^cases

being

similar. Fix 3

general planes M, N,

^1~.

Take limits

W, X, D, respectively

of chains of

[(a +

^b

+ 1)/4]7 [(a -f-

^{b -}

1)/4],

^and

2, conics,

^with

.M~, X, Dred c R,

W D

intersecting transversally Q,

^{W W X ~J} D limit of a

family

of chains of

conics,

^card

(D

ⁿ

Q

ⁿ

M)

⁼

2,

^card

(D nQnN)

^=2.

card

(X

ⁿ

Q

ⁿ

ll ) =1.

^Set

Any

^{forms of}

type (a, b) vanishing

^on

A,

^{vanishes on ac}

+

^b

+

¹

points

^on

M’,

^{hence on ~1’.}

Any

^{form of}

type (a - 1, b - 1) vanishing

^{on the}

points

^of

n vanishes on a

+

^{b - 1}

points

^of

N’,

^{hence on N’. We}

reduce to an assertion about forms of

type (a - 2, b - 2) (in

^which

we have to consider also the 4

points

^{in D r1}

(QB(.M’ ~J N’))~.

^We

continue in the same way, ^never

adding

^curves

intersecting

^{M’ U}

N’ ;

for the next

step

^{we take .R as first}

working plane.

At the end in

case

(i)

^{we reduce to the case a = 3 or 4}

(plus

^{a few}

points),

^{in case}

(ii)

to cases with a 4. Consider for instance the case a = b ⁼ 3 and no

point

left from the

previous

construction

(the

^worst

case).

^{Take 3}

general

^{smooth curves}

L, L’,

^{L" of}

type (1, 1 )

^on

Q,

^{and let}

H, H’,

^H"

the

planes they

span. The

following

^{chain of}

conics solves our

problem.

^A

(resp. A’ )

^{is a}

sufficiently general

^conic

in g

(resp..g’ ),

^{A" is a}

sufficiently general

^{conic in .~"}

containing

^a

point

^of

L,

^{B is a conic}

containing

²

points

^{of L and a}

point

^of

L’,

but not

intersecting

^L".

The aim of this section is the

proof

^{of the}

following

^lemma.

LEMMA 2.2. Fix

integers n, x,

^with

n &#x3E; 29, 0 c a c 2~~ - 2.

There

exists

( Y, Z, T)

^{with chain of}

conics,

T bamboo of

degree a, ry,3(n) surjective, deg ( Y) ~ r(n, 0, 3) -

^{9 - n/6}

PROOF. Let 8 be the maximal

integer

^{with s = n mod}

(4),

say n ^{= s}

+ 4t, r(s, 0, 3) - 3 (r(n, ^{0, 3) -} a)/2 - ^{(n -} s)13. By [4], 3.1,

^{there is a} bamboo E E

P3, deg (.E) = r(s, 0, 3) - 2,

^with

surjective.

^{If a =}

0,

^set

I’ : _ .~,

^{T = 0. If a} &#x3E;

0,

^take

F c E, deg (F) = deg (E) - ~ ,

^y F union of two

disjoint bamboos A, T, deg (T) _

^{~c. Then we fix a smooth}

quadric Q

^{and we}

apply

^{the so}

called Horace’s method

(introduced

ⁱⁿ

[8])

^2t times. At the odd

(resp. even) steps

^{we add in}

Q

^{lines of}

type (1, 0 ) (resp. (0, 1)).

^We

order the lines

Ll’’’.’ Ldeg(E)

of E in such ^a way that

Li

^r1 if and

only - j

^{C 2. Just to fix} the notations ^we assume s ⁼

61~,

7~

integer (which, together

^{with s =}

6k ~ 5,

^{is the worst}

case).

^We

add in. Q

the union U of

4k+

^{2 =}

r(6k -f- 2, 0, 3)

^-

(6k -E- 2, 0, 3) -

¹

lines

A i , I i 4k + 2,

^of

type (1, 0 )

^with

A

ⁱ

intersecting L2i-l

^for

every i. We claim that

--~- 2)

^is

surj ective

^{f or}

general

^F.

To prove the

claim,

^{it is} sufficient to find S C

P3,

in

P31Q, S"

r) S’ = f~ S"

general

ⁱⁿ

Q.

^Fix

f

^E

⁺ 2)).

By

^{2.1 and the}

generality

^of

~",

^for

general

^{~’ we} may ^assume

flQ

^{= 0. Thus}

^f

^{is divided}

by

^the

equation 9

^of

Q.

^{Since vanishes}

on I’ we have ⁼

0,

hence the claim is

proved.

^{Then we deform}

the lines

A

^{to hues}

A

^{with the}

following

rule. Let

U’

be the union of the lines We assume that

U’

intersects

transversally Q. Ai

^in-

tersects

Lj

^{if and}

only

^if

^Ai

intersects

Li . Ai intersects Q

^{at a}

point

on a line

B1

^of

type (0, 1) intersecting ^{L2 .} Inductively,

^we

impose

^that

Q, j

&#x3E;

1,

^{has a}

point

^{on a line}

Bi

^of

type (0, 1) intersecting L2

a

and a

point

^{on 4he line}

intersecting L2i-2

^and

A’-I.

^{The lines}

^Bi

are called

« good»

secants to .F’ U U’. Then we

repeat

the Horace’s method in the

following

way. To F ^u U’ ^we add in

Q4k +

^{4 =}

=

r(6k + 4, 0, ~) - r(6k + 2, 0, 3)

^lines

Bi

^of

type (0, 1),

among them the

« good »

^secants

previously constructed, B6k+S

^{linked to}

L1, B6k+4

linked to Then we continue

(after smoothing

the reducible conics

obtained,

if you

prefer).

^{In the}

step

^{from m - 2 to m we add}

to a curve

X (m - 2)

ⁱⁿ

Qr(m, 0, 3 ) - r(m - 2, 0, 3)

^{lines if}

q(m, 0, 3)

&#x3E;

~ q(m - 2, 0, 3), r(m, 0, 3) - r(m - 2, 0, 3) -

^{1 lines if}

q(m, 0, 3) q(m - 2, 0, 3) (i.e.

^{if m - 0 or 5 mod}

(6)).

In the odd

step

^from

m - 2 to m we add a certain

number,

say x, of lines of

type (1, 0),

and creates x

« good»

secants. In the even

step

^{from m to m}

+

²

we add to

X (m)

^{the x}

« good»

^secants creates in the

previous step

and one or two lines

(always

^of

^type (0,1 ))

linked either to a free conic of or to a free line in a bamboo of

X(m).

^{We use} that after 6

steps

the lines added in the 3 even

steps

^are

exactly

^{4 more than the}

lines added in the 3 odd

steps.

^This

explain

^{the term c -}

(n - s ) /3 »

in the choice of s. ·

Consider the

following

^assertion

T(n,

a,

b),

defined for all

integers

n,

a, b. T(n,

a,

b ) :

^{There is}

( Y, Z, ^{T )}

^with

Z chain of a

conics,

T bamboo of

degree b,

^{and with}

surjective.

In the

following section,

^{for the}

proof

of theorems

1,

^{2 we will}

need the assertion

T(n,

cc,

b)

^for the values of n, a,

b,

^{listed in 2.3.}

LEMMA 2.3. - The assertion

T(n,

a,

b)

^{is true if}

(n,

a,

b)

^{has one of}

the

following

^values:

(2, 1, 0), (2, 0, 2 ), (3, 2, 0 ), (3, 2, 1), (4, 3, 0 ),

(4, 2, 2 ), (5, 4, 0), (5, 3, 2 ), (6, 4, 2 ), (6, 3, 4), (7, 2, 8), (7, 5, 3 ), (8, 3, 10 ),

(8, 5, 5), (9, 9, 2 ),

2

(9, 7, 4),

²

(10, 8, 5 ), (10, ~, 12 ), (11, 9, 7), (11, 5, 14),

(12, 11, 6), (12, 7, 15), (13, 14, 4), (14, 14, 8), (1~, 16, 9), (16, 18, 9),

(17, 22, 6), (18, 16, 24), (18, 22, 11), (19, 18, 24), (19, 25, 11), (20, 27, 12),

(21, 20, 33), (21, 32, 8), (22, 33, 12), (22, 25, 29), (23, 35,15), (23, 27, 30),

(24, 29, 13), (24, 30, 32), (25, 45, 8), (26, 45, 15), (27, 48, 17), (28, 52, 16), (29, 59, 10), (30, 48, 41).

Sketch of

proof.

^{The cases with n 24 can be done}

using

^several

times the Horace’s costruction

applied

^{not to}

quadrics

^{but to}

planes:

it is

easier;

^{if n}

10,

^{we do not} need any

nilpotent,

if n &#x3E; ^{9 we use}

nilpotents

^{as in}

[8]; only

^{the cases with n} &#x3E; 21 are more

difficult

y however

they

^{can be} handle also

using quadrics

^{as in the}

proof

^{of 2.2.}

If n &#x3E;

23,

^the

proof

^{of 2.2 works}

verbatim,

^and

gives

^indeed

stronger results;

^for

(24, 39,13 )

^start

taking

^{in the}

proof

^{of 2.2 s =}

12;

^{for the}

remaining (n,

a,

b)

start from s = n - 8. ·

3. In this section we show how to

modify

^the

proofs

ⁱⁿ

[4],

^to

prove theorems

1,

^{2. The}

proof

^{of the case « P4 »}

given

ⁱⁿ

[4], § 8,

cannot be

adapted,

^but the results proven here

in §

^{2 are sufficient}

to prove this ^case and the inductive assertions of

[4]

needed for the

proofs

ⁱⁿ

PN,

N &#x3E; 5. We will use the numbers

y(k, N),

^{... , intro-}

duced

in §

^0.

Consider the

following

assertions:

Y(k, N), k

&#x3E;

0, n

&#x3E; 3: there exists a chain Y of

y(k, N)

^smooth

conics in PN with

surjective;

if either k &#x3E; 6 or k &#x3E;

2,

^N &#x3E;

4,

or N &#x3E;

6,

^{there is such a} Y which is contained in an

integral hyper-

surface of

degree

^k.

Z(k, N), k

&#x3E;

0,

^N &#x3E; 3: there exists a chain Y of

c(k, N) -~- k -

¹

smooth conics in PN with

injective.

W(k,a,N,j), k&#x3E;2, N&#x3E;3,

for every

subset S c PN,

in linear

general position,,

and every

A,

^{B E}

~’, B,

^{there is a curve} Y c PN such that:

(a)

^{Y n S = -}

~A., B~,

^is

surjective

^{and a}

general hypersurface

^of

degree k containing

^is

irreducible;

H(k, N), k

&#x3E;

0,

^N &#x3E; 3: there exists a curve

such that:

(b)

^{T is a bamboo of}

degree 2j(k, N) intersecting Z exactly

at a

point,

say

P,

^and

quasi-transversally;

^{P is a}

point

in a final line of

T;

(e) rY,N(k)

^is

^bijective.

W (k,

a,

N, j )

^and

H(k, ^N)

^are

slight

modifications of the assertions of

[4]

^{with the same name. In}

[4] Y(k, N)

^and

Z(k, N)

^were

proved

for N &#x3E; 4. The same method

(Horace’s

construction

using

^a

hyper- plane) gives Y( k, 4), Z(k, 4), using

^{2.2 if k} &#x3E; 30

(plus

^{a numerical}

lemma:

c 2e(k, 4)

^{- 2} &#x3E;

r(k, 0, 3)/2 + (k2/6) +

^{10k if k} &#x3E; 30 » whose

proof

^{is left to the}

reader), using

^{2.3 if k 31.}

In the same way ^we

get

^{the «new»} assertions

W(k,

ac,

4, j), H(k, 4).

Then the

proofs

^{of the « new»}

W(k,

a,

N, j), H(k, N),

N &#x3E;

4,

^{are done}

by

^{induction as in}

[4];

^{the cases with N = 4}

simplify

the discussion of the cases with low k for N ⁼ 5

given

ⁱⁿ

[4],

^6.4. Then theorem 1 is

proved

ⁱⁿ

3,

^{in the same} way the

corresponding

^{theorem is}

proved

^for

PN ,

N &#x3E;

4,

ⁱⁿ

[4],

^end

of §

^{7. The same}

proof

^{works for}

theorem

0.2, although

^a

simpler

^one could be done in this case,

adding

in a

hyperplane

irreducible

hyperelliptic

^curves.

4. After this paper ^was

typed,

^{we read}

[6].

^{It is}

elementary

^to

show how the results of

[6],

^th.

4, 5,

^about

sygygies

^of

general

^canonical

curves can be

adapted

^to

give

results about

syzygies

^of

general k-gonal

^curves for suitable k. We have:

PROPOSITION 4.1. Let X be a

non-hyperlliptic k-gonal

genus n

curves. Assume

g~,2(X)

^{= 0 for an}

^integer

^{p with}

^{1 p n -}

³

p c k - 3.

^Then

(a)

^{If C is a}

general k-gonal

^curve of genus n

+

p

+ 1,

^then

(b)

^{If C is a}

general k-gonal

^curve of genus m, where

For the

proof

^of

4.1,

^{it is} sufficient to take in the

proof

^of

[6],

^th.

4,

as divisor q,

+

q2

+

^...

-~-

qrp+2 ^a divisor contained in

a gk (strictly

contained

by

^the

assumption k

&#x3E; p

+ 2)

^{and as}

smoothing

^{of X U Y}

an admissible k-cover. Then from 4.1 we

get

verbatim the

following

improved

^version

of [6],

^{th. 5:}

PROPOSITION 4.2. Let C be a

general k-gonal

^curve of genus g.

(a) K2,2(C)

^{= 0 if}

g &#x3E; 7

and either k &#x3E; 5 or

g -1,

^{2 mod}

(3)

^and

k = 5.

(b ) ~3,2(5)

^{- 0 if}

g &#x3E; 9

and either k &#x3E; 6 or 2 mod

(4)

^and

k = 6.

(c) KI,2(C)

^{= 0 if}

6, and g =1, 2

^mod

(5).

REFERENCES

[1] ^{E. BALLICO - PH.} ELLIA, ^{Generic curves}

of

^small genus in P3 are

of

^max-

imal rank, Math. Ann., 264

(1983),

pp. 211-225.

[2] ^E. BALLICO - PH. ELLIA, ^On

degeneration of projective

^{curves, in}

Algebraic Geometry-Open

^{Problems, Proc.} Conf., Ravello, 1982,

Springer

^Lect.

Notes Math. 997 (1983), pp. 1-15.

[3] E. BALLICO - PH. ELLIA, ^{On the}

hypersurfaces containing

^a

general

^pro-

jective

^curve,

Compositio

Math., 60 (1986), pp. ^85-95.

[4] E. BALLICO - PH. ELLIA, Postulation

of general

^{canonical curves in}

PN,

N ~ 4, ^Boll. U.M.I., ^{Sez. D} (1986), pp. 103-133.

[5] ^M.-C. CHANG, Postulation

of

canonical curves in

P3,

^Math. Ann., ²⁷⁴

(1986),

pp. 27-30.

[6] ^L. EIN, ^{A remark on} the

syzygies of

^the

generic

^{canonical curves, J. Diff.}

Geom., ²⁶ (1987), pp. 361-365.

[7] ^J. HARRIS - D. MUMFORD, On the Kodaira dimension

of

the moduli space

of

curves, Invent. Math., 67 (1982), pp. ^23-86.

[8] ^A. HIRSCHOWITZ, Sur la

postulation générique

des courbes rationnelles, Acta Math., 146 (1981), pp. 209-230.

[9] ^A. TANNENBAUM,

Deformation of

space curves, Arch. Math., ³⁴

(1980),

pp. 37-42.

Manoscritto pervenuto in redazione il 16 ottobre 1987.